Their relation to \(T\bar{T}\) -like deformations was first pointed out in [1]}, [2]}. Also in the spin chain case deformations using `internal' symmetries have been explored [3]} and, importantly for the present paper, a generalization to systems with open boundaries exists [4]}. Due to the discrete nature of the spin chain model, it is yet unknown how to define the `real' \(T\bar{T}\)  deformation for lattice models, which requires the conserved momentum current. Bilinear deformations of integrable non-relativistic quantum many-body systems such as the 1d Bose gas (non-linear Schrödinger model) have been studied in [5]}, [6]}, [7]}, [8]}, [9]}, [10]}. For these models, the \(T\bar{T}\)  deformation can be defined and it was found that the deformation has the effect of changing the length of fundamental particles of the model. The simplest deformation that changes the length of particles is the hard rod deformation defined in [5]}, [6]}. It turns out that the hard rod deformations can be defined for a wide range of models including lattice models. The hard rod deformation of spin chain models can be identified with the constrained XXZ [13]}, [14]}, [15]} and folded XXZ spin chain [16]}, [17]}, [18]}, [19]}, [20]}, which have recently received renewed interest from different perspectives [21]}, [22]}, [23]}.
